Let D(p) be the demand for a good at price p. The revenue function is the amount of money gained by the producer when fulfills the demand, namely R(p) = pD(p). 1. Let E(p) be the elasticity of D(p). Compute R'(p) in terms of D(p) and E(p) only. 2. Recall that, in general, the elasticity of the demand is negative, that is, E(p) < 0. We assume that this is the case from now on. Now, If the demand is elastic, what is the sign of R'(p)? What happens to the revenue if we slightly increase the price? What if the demand is inelastic? 3. According to the previous question, if the revenue is maximal at p, can the demand be elastic at p? Inelastic? What can you conclude?
Let D(p) be the demand for a good at price p. The revenue function is the amount of money gained by the producer when fulfills the demand, namely R(p) = pD(p). 1. Let E(p) be the elasticity of D(p). Compute R'(p) in terms of D(p) and E(p) only. 2. Recall that, in general, the elasticity of the demand is negative, that is, E(p) < 0. We assume that this is the case from now on. Now, If the demand is elastic, what is the sign of R'(p)? What happens to the revenue if we slightly increase the price? What if the demand is inelastic? 3. According to the previous question, if the revenue is maximal at p, can the demand be elastic at p? Inelastic? What can you conclude?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Exercise I: Elasticity and Revenue I
**Prerequisites**: Elasticity
Let \( D(p) \) be the demand for a good at price \( p \). The revenue function is the amount of money gained by the producer when it fulfills the demand, namely
\[ R(p) = pD(p). \]
1. Let \( E(p) \) be the elasticity of \( D(p) \). Compute \( R'(p) \) in terms of \( D(p) \) and \( E(p) \) only.
2. Recall that, in general, the elasticity of the demand is negative, that is, \( E(p) < 0 \). We assume that this is the case from now on. Now, if the demand is elastic, what is the sign of \( R'(p) \)? What happens to the revenue if we slightly increase the price? What if the demand is inelastic?
3. According to the previous question, if the revenue is maximal at \( p \), can the demand be elastic at \( p \)? Inelastic? What can you conclude?
4. We now study a specific case. Assume that the demand for broccoli is given by \( D(p) = 5000 - 500p^2 \), where \( D(p) \) is in tons and \( p \) is the price of a pound of broccoli.
(a) Compute the elasticity of the demand as a function of price.
(b) Find \( p \) such that the demand at \( p \) is unit elastic. Give the exact value and an approximation with two significant digits. What does this price mean in terms of revenue?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d8618e4-e230-4c20-b760-ebcb33f527d3%2Fa249764f-775f-4bc0-848b-878e8dfc6561%2Ferdrybq_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise I: Elasticity and Revenue I
**Prerequisites**: Elasticity
Let \( D(p) \) be the demand for a good at price \( p \). The revenue function is the amount of money gained by the producer when it fulfills the demand, namely
\[ R(p) = pD(p). \]
1. Let \( E(p) \) be the elasticity of \( D(p) \). Compute \( R'(p) \) in terms of \( D(p) \) and \( E(p) \) only.
2. Recall that, in general, the elasticity of the demand is negative, that is, \( E(p) < 0 \). We assume that this is the case from now on. Now, if the demand is elastic, what is the sign of \( R'(p) \)? What happens to the revenue if we slightly increase the price? What if the demand is inelastic?
3. According to the previous question, if the revenue is maximal at \( p \), can the demand be elastic at \( p \)? Inelastic? What can you conclude?
4. We now study a specific case. Assume that the demand for broccoli is given by \( D(p) = 5000 - 500p^2 \), where \( D(p) \) is in tons and \( p \) is the price of a pound of broccoli.
(a) Compute the elasticity of the demand as a function of price.
(b) Find \( p \) such that the demand at \( p \) is unit elastic. Give the exact value and an approximation with two significant digits. What does this price mean in terms of revenue?
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